We present all the Target of Opportunity (ToO) sub-millimeter observations of GRB 030329 taken by the Sub-millimetre Common-User Bolometer Array (SCUBA) on the James Clerk Maxwell Telescope (JCMT). This was by far the brightest sub-millimeter afterglow seen to date. The flux density at 850 $\mu$m was approximately constant up to a break that took place $\sim $7 days after the burst. This was consistent with being a jet break. The 850 $\mu$m results agree with those at longer wavelengths that show a brighter flux $\sim $7 days after the burst, right at the time of the break. No short-lived large-scale brightenings were detected in the sub-millimeter light curve. However, the 850 $\mu$m light curve may have had a drop $\la$16 days after the burst. The peak of the afterglow emission was at ~90 GHz in the days before the break in the light curve. A simple modeling is consistent with the spectral indices remaining the same as the afterglow evolved, with the breaks in the spectrum moving to longer wavelengths at later times and the flux at the peak falling. No significant sub-millimeter emission was detected from the host galaxy.